{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T17:04:17Z","timestamp":1775667857485,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T00:00:00Z","timestamp":1750118400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Key R&amp;D Program of China","award":["2022YFB2703500"],"award-info":[{"award-number":["2022YFB2703500"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>As a core strategy for carbon emission reduction, carbon trading plays a critical role in policy guidance and market stability. Accurate forecasting of carbon prices is essential, yet remains challenging due to the nonlinear, non-stationary, noisy, and uncertain nature of carbon price time series. To address this, this paper proposes a novel hybrid deep learning framework that integrates dual-mode decomposition and a TKMixer-BiGRU-SA model for carbon price prediction. First, external variables with high correlation to carbon prices are identified through correlation analysis and incorporated as inputs. Then, the carbon price series is decomposed using Variational Mode Decomposition (VMD) and Empirical Wavelet Transform (EWT) to extract multi-scale features embedded in the original data. The core prediction model, TKMixer-BiGRU-SA Net, comprises three integrated branches: the first processes the raw carbon price and highly relevant external time series, and the second and third process multi-scale components obtained from VMD and EWT, respectively. The proposed model embeds Kolmogorov\u2013Arnold Networks (KANs) into the Time-Series Mixer (TSMixer) module, replacing the conventional time-mapping layer to form the TKMixer module. Each branch alternately applies the TKMixer along the temporal and feature-channel dimensions to capture dependencies across time steps and variables. Hierarchical nonlinear transformations enhance higher-order feature interactions and improve nonlinear modeling capability. Additionally, the BiGRU component captures bidirectional long-term dependencies, while the Self-Attention (SA) mechanism adaptively weights critical features for integrated prediction. This architecture is designed to uncover global fluctuation patterns in carbon prices, multi-scale component behaviors, and external factor correlations, thereby enabling autonomous learning and the prediction of complex non-stationary and nonlinear price dynamics. Empirical evaluations using data from the EU Emission Allowance (EUA) and Hubei Emission Allowance (HBEA) demonstrate the model\u2019s high accuracy in both single-step and multi-step forecasting tasks. For example, the eMAPE of EUA predictions for 1\u20134 step forecasts are 0.2081%, 0.5660%, 0.8293%, and 1.1063%, respectively\u2014outperforming benchmark models and confirming the proposed method\u2019s effectiveness and robustness. This study provides a novel approach to carbon price forecasting with practical implications for market regulation and decision-making.<\/jats:p>","DOI":"10.3390\/sym17060962","type":"journal-article","created":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T05:52:47Z","timestamp":1750139567000},"page":"962","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Carbon Price Forecasting Using a Hybrid Deep Learning Model: TKMixer-BiGRU-SA"],"prefix":"10.3390","volume":"17","author":[{"given":"Yuhong","family":"Li","sequence":"first","affiliation":[{"name":"Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nan","family":"Yang","sequence":"additional","affiliation":[{"name":"Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guihong","family":"Bi","sequence":"additional","affiliation":[{"name":"Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shiyu","family":"Chen","sequence":"additional","affiliation":[{"name":"Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhao","family":"Luo","sequence":"additional","affiliation":[{"name":"Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xin","family":"Shen","sequence":"additional","affiliation":[{"name":"Measurement Center of Yunnan Power Grid Co., Ltd., Kunming 650051, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1016\/S0140-6736(06)68079-3","article-title":"Climate change and human health: Present and future risks","volume":"367","author":"McMichael","year":"2006","journal-title":"Lancet"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1673","DOI":"10.1007\/s10668-022-02780-y","article-title":"Factors driving CO2 emissions: The role of energy transition and brain drain","volume":"26","author":"Kazemzadeh","year":"2024","journal-title":"Environ. 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